Question

Consider the following L.P.P.
Maximize $Z=3x+2y$
Subject to the constraints
$x+2y\le 10$
$3x+y\le 15$
$x,y\ge 0$
(a) Draw its feasible region.
(b) Find the corner points of the feasible region.
(c) Find the maximum value of $Z$.

Answer 1

(a)

$Z=3x+2y$ .......... $\left(i\right)$

subject to the constraints

$x+2y\le 10$

$3x+y\le 15$

Convert these inequalities into equations

$x+2y=10$ ......... $\left(ii\right)$

$3x+y=15$ ......... $\left(iii\right)$

From $\left(ii\right)$, we get

$x=0⟹y=5$ and $y=0$ when $x=10$

So, the points $(0,5)$ and $(10,0)$ lie on the line given in $\left(ii\right)$

From $\left(iii\right)$, we get the points

$(0,15)$ and $(5,0)$

Let's plot these point and we get the graph in which, shaded part shows the feasible region.

(b)

Lines $\left(ii\right)$ and $\left(iii\right)$ intersect at $(4,3)$ and other corner points of the region are $(0,5),(5,0)$ and $(0,0)$.

(c)

To find the maximum value of $z$, we need to find the value of $z$ at the corner points

Corner points $z=3x+2y$

$(0,0)$ $0$

$(5,0)$ $15$

$(0,5)$ $10$

$(4,3)$ $18$

Thus, $z$ is maximum at $(4,3)$ and its maximum value is $18$.